generalized ordered space
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2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Ai-Jun Xu

AbstractIn this paper, we show that any generalized ordered space X is monotonically (countably) metacompact if and only if the subspace X - { x } is monotonically (countably) metacompact for every point x of X and monotone (countable) metacompact property is hereditary with respect to convex (open) subsets in generalized ordered spaces. In addition, we show the equivalence of two questions posed by H.R. Bennett, K.P. Hart and D.J. Lutzer.


2015 ◽  
Vol 3 (1) ◽  
Author(s):  
A. Dorantes-Aldama ◽  
R. Rojas-Hernández ◽  
Á. Tamariz-Mascarúa

AbstractIn the set of compactifications of X we consider the partial pre-order defined by (W, h) ≤X (Z, g) if there is a continuous function f : Z ⇢ W, such that (f ∘ g)(x) = h(x) for every x ∈ X. Two elements (W, h) and (Z, g) of K(X) are equivalent, (W, h) ≡X (Z, g), if there is a homeomorphism h : W ! Z such that (f ∘ g)(x) = h(x) for every x ∈ X. We denote by K(X) the upper semilattice of classes of equivalence of compactifications of X defined by ≤X and ≡X. We analyze in this article K(Cp(X, Y)) where Cp(X, Y) is the space of continuous functions from X to Y with the topology inherited from the Tychonoff product space YX. We write Cp(X) instead of Cp(X, R).We prove that for a first countable space Y, K(Cp(X, Y)) is not a lattice if any of the following cases happen:(a) Y is not locally compact,(b) X has only one non isolated point and Y is not compact.Furthermore, K(Cp(X)) is not a lattice when X satisfies one of the following properties:(i) X has a non-isolated point with countable character,(ii) X is not pseudocompact,(iii) X is infinite, pseudocompact and Cp(X) is normal,(iv) X is an infinite generalized ordered space.Moreover, K(Cp(X)) is not a lattice when X is an infinite Corson compact space, and for every space X, K(Cp(Cp(X))) is not a lattice. Finally, we list some unsolved problems.


2011 ◽  
Vol 83 (3) ◽  
pp. 463-469 ◽  
Author(s):  
YIN-ZHU GAO ◽  
WEI-XUE SHI

AbstractFor any generalized ordered space X with the underlying linearly ordered topological space Xu, let X* be the minimal closed linearly ordered extension of X and $\tilde {X}$ be the minimal dense linearly ordered extension of X. The following results are obtained. (1)The projection mapping π:X*→X, π(〈x,i〉)=x, is closed.(2)The projection mapping $\phi : \tilde {X} \rightarrow X_u$, ϕ(〈x,i〉)=x, is closed.(3)X* is a monotone D-space if and only if X is a monotone D-space.(4)$\tilde {X}$ is a monotone D-space if and only if Xu is a monotone D-space.(5)For the Michael line M, $\tilde {M}$ is a paracompact p-space, but not continuously Urysohn.


2003 ◽  
Vol 4 (1) ◽  
pp. 71
Author(s):  
Masami Hosobuchi

<p>The notion of the S<sub>δ</sub>-diagonal was introduced by H. R. Bennett to study the quasi-developability of linearly ordered spaces. In an earlier paper, we obtained a characterization of topological spaces with an S<sub>δ</sub>-diagonal and we showed that the S<sub>δ</sub>-diagonal property is stronger than the quasi-G<sub>δ</sub>-diagonal -diagonal property. In this paper, we define a dense S<sub>δ</sub>-diagonal of a space and show that two linearly ordered extensions of a generalized ordered space X have dense S<sub>δ</sub>-diagonals if the sets of right and left looking points are countable.</p>


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